To clear everything (this will include cookies and passwords), swipe left and tap the gear button to open the Preferences. Then scroll down and tap the Clear private data button. If you want to only clear your history, you can install the Cleary add-on, which will give you a finer-grained ability to remove items from the same preference.

Do I understand that there is no way to JUST clear the Awesome screen? As I noted above, Cleary cleared the History page, but the Awesome screen still has all the websites that I wanted to get rid of. I would hate to clear out my cookies and passwords, but is that the only way to actually clear the Awesome screen? Do I really need to clear all the private data?

you should normally have a bolt icon or some kind of mechanical tool on your phone that allows you to select advanced options. Press that button and see if there is the option to clear the history from there. Do it on the Awesome Screen.

Thanks for your additional help. Perhaps the problem was that I'm using a Samsung Galaxy Tab 10.1 and not a phone. Anyway, I WAS able to finally get the sites off the Awesome screen by first removing the star (bookmark?) that was associated with those sites and then using Cleary to clear the History again. That finally did what I wanted to do.